Inspired by playing the excellent Sumaze! 2 game from MEI, here is an investigation that aims to provide some purposeful practice on decimals. The aim is to provide an accessible entry point for all learners with opportunities for depth through generalisation. This slide presentation steps through it although exactly how you move from one part to the next will, of course, depend on the class.

I have included solutions in this spreadsheet although I would be hesitant to display them in this form, as I would prefer that the results are found and discussed as we go along rather than just revealing them at the end.

It was a great pleasure to host Rob Eastaway at the London branch ATM/MA meeting this morning. His theme was Arithmetic, and how some techniques are almost becoming a lost art. There was so much energy in this session, the room was positively buzzing with pencils and pens scratching away! We covered so much ground in two and a half hours, I’m not going to attempt to write about everything but I am going to pick the thing that resonated most with me.

A number of techniques we explored were about getting exact answers, but this section was as rough as you like! Rob introduced us to his idea of “Zequals“. When teaching rounding, I always enjoy introducing my students to the “approximately equals” sign, ≈. I hadn’t really considered how this symbol, on its own, doesn’t give the complete story. All of these statements are true…

7.3 ≈ 7
7.3 ≈ 10
and even 7.3 ≈ 7.4

but they don’t give an explanation of what you have actually done to the 7.3 and in the last example here, it really would require quite a bit of explanation!

So Rob proposes “Zequals” which has a precise method. It looks like this:

and it involves rounding the numbers you calculate with AND the result to 1 significant figure.

The Numberphile video explains in more detail here. An interesting question to ask might be, what calculation would give the biggest discrepancy between the accurate calculation and the Zequals calculation? And what would the % error be? The blog post explains this beautiful graphical representation of that % error, which turns out to be a fractal.

Now, to be honest, I would be hesitant to “teach” non-standard notation and methods as part of the regular timetable of maths. There is already so much to learn and time is so precious, why would I take a lesson explaining something they are unlikely to ever encounter again in this form? But dismissing it on that basis, misses the point, I feel.

Estimation as a topic features in a fairly minor way at GCSE but is a critically important skill in many jobs and life in general. There was some discussion amongst the attendees this morning that as students progress through KS3 to KS4 and A-level they become more and more reliant on their calculator. With the demise of the C1 paper, there is no longer a requirement for a non-calc paper at A level which is inevitably going to mean that our students will get weaker at this skill rather than stronger. This seems like a real mismatch between our education system and the requirements of employees and our broader society.

An idea which might help is to explicitly teach estimation as a technique to be employed when doing calculations with large numbers or decimals. Typically these types of calculations would involve some sort of “ignoring” the decimal point or the zeros, doing the calculation, and then “putting it back”

Now I am not claiming that this is a more efficient or reliable method. It does depend to a certain extent on the examples chosen and “counting the decimal places” is a method that will always work. But I feel that the approximation step helps with number sense: the idea that 20 × 0.3 is a bit less than half of 20 so must be 6 is really valuable for life beyond exams. It provides an opportunity to practise these skills, practice which I believe our students currently have precious little of.

This little gadget on Mathisfun.com is so illuminating for teaching decimals. I always get a positive response from students when I show it.

I think it can be used it in a number of ways:

Predict what happens when I zoom in. As you zoom in, first the tick marks appear and then at the next level of zoom, numbers start appearing. It’s a great way of “bridging” from the familiar to the new. A number line is likely to be a familiar concept. However, what is “in between” 0 and 1? An intentionally ambiguous question. Students are likely to say “half” or 0.5. How else can we show half? What other numbers are in between 0 and 1?

Why are there 10 ticks between 0 and 1? We have divided 1 into 10 equal parts, what is each part. How else can we represent 1/10?

From there we “zoom” into the next level of hundredths

This table might be useful as some practice to relate fractions to decimals. I would love to hear some comments on this.

This is what came to mind when joining Derek Ball’s session at the ATM conference this week, entitled “Recurring Decimals”.

In fact, we were going the other way: converting fractions to recurring decimals, specifically looking at fractions where the denominator is prime. It was a fascinating session, great for deepening subject knowledge. This blog post is my attempt to reflect on what I learned and my thoughts about how I might use this in the classroom, probably Year 10 or 11, but really any group that is confident with bus stop division could investigate this.

I was already aware of some pretty cool things that happen with sevenths, mainly from Don Steward’s blog:

Why should this happen? Why do we only see these six digits with sevenths? The process of bus stop division helps us see why and this is where I feel I would start with a class. This is good practice of a technique that should be secure but often isn’t in Year 9 / 10. It helps learn the seven times table and I don’t think it is too tedious to ask students to perform these six calculations manually:

Some learners might want to find 2/7 by doubling 1/7. And then maybe find 3/7 by adding 1/7 to 2/7 and so on. Even if they stick with the bus stop division, they will find that they are essentially doing the same six calculations:

The ones digit is always zero and the tens digit must be less than 7 so there are only these 6 options. Can we extend this rule to other fractions with denominator less than 10? Of course, all others except 3, 6 and 9 will terminate – why is this?

There are also some interesting things to notice about the 1/7 “wheel” before moving onto higher primes. I won’t spoil it for you, but suggest you include the fractions around the outside of the wheel to spot some of the patterns.

We then moved onto looking at 13ths. 11th are interesting too for different reasons, so I can see why we went onto 13th because there are some surprising relationships between 7th and 13ths. At this point, we started using calculators and I would do the same with a class. Or even better, open a spreadsheet, which I am always keen to do!

I’m not sure of the value of kids typing 12 calculations into their calculators, so I might give this image above as a print out for them to write on. Hopefully they will soon spot that there are two sets of recurring digits, i.e. 076923 in 1/13 + 5 others and 153846 in 2/13 + 5 others and which, this time can be written as two wheels:

Ideally at this point you’ll have the class hooked and they would be asking all sorts of questions. Well, maybe enough of them to get everyone else thinking. I would try really hard to encourage the students to come up with these questions. This is what I hope they will ask, but I hope they will also ask questions I hadn’t thought of, something which is a really special moment in any lesson.

Do we see the same patterns in the sevenths wheel as we do in the thirteenths wheel?

Is there something special about 6 points around the wheel?

Why are there 2 wheels? Are there any patterns across the two wheels?

What about other fractions, do they have multiple wheels?

On the image above, I have shown that as you move clockwise around the wheel, you are effectively multiplying by 10. This is obvious going from 1/13 to 10/13 but actually occurs for all other hops if we ignore the whole number part. i.e.

So if go 3 hops we have multiplied by 1000. I think this goes some way to explaining why the fractions that are opposite each other must sum to 1.

The next prime fraction: 17ths recur after its maximum of 16 digits, so effectively we have one, rather large wheel. Unfortunately, Excel gives up after 16 decimal places (please let me know if there is a way around this) but you can still see some patterns in here:

Beyond that, we found that the following fractions recur after the maximum number of digits so have only one wheel: 3,7,17,19,23,43,59,61 – someone had a much better calculator than me that showed 32 digits!

Other fractions worth exploring are:

31ths – contain 2 wheels of 15

37ths – contain 12 wheels of 3

41th – contain 8 wheels of 5

With the fractions that recur after a relatively few numbers of digits, we can find factors.

Because 1/41 is a decimal that recurs after only 5 digits, if follows that 41 must be a factor of 99999.

And in the larger wheels there are all sorts of patterns – pentagons and triangles in the wheel of 31ths apparently.

I’m not sure how far I’d go with a class, there could be several lessons worth in here. It would depend on how they responded, of course. Maybe 13ths would be sufficient unless…

This post captures only some of what we worked on in this session and highlights to me the depth of subject knowledge that can be gained by attending the ATM conference. Other sessions were more pedagogical in nature, but this one was pure fascinating mathematics and I was grateful to be surrounded by so many knowledgable and friendly people!

An idea for a mixed attainment class that came to me about 5 minutes before a lesson today:

3.4 – 3.04

5.2 – 5.02

7.8 – 7.08

8.2 – 8.02

Find other questions like this. (The “weakest” student in the class told me the pattern before I’d even finished writing the fourth question on the board.)

What do you notice? Why is the answer to Q2 the same as the answer to Q4?

Can you create a question with 0.54 as an answer? How many different answers are there to these types of questions?

Then:

5.7 – 5.007

8.3 – 8.003

6.4 – 6.004

etc.

These are more tricky and test the skills of column subtraction, something that should be secure by Year 7 but may not be. Maybe an opportunity for collaboration amongst students to show how.

And then finally, try these two calculations. Which is easier and why?

7 – 1.392

6.999 – 1.391

Show on a number line why this works and then try some more. I think these questions are interesting to explore. But I would hesitate to recommend it as a must-do method to solve e.g 8-2.5687. Whether or not it is easier to turn it into 7.999-2.5686 or not is an interesting discussion and one which I would want my students to form their own opinion on, not be too swayed by mine.